Textbook Question
Find the derivative the following ways:
Using the Product Rule or the Quotient Rule. Simplify your result.
f(x) = (x - 1)(3x + 4)
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 8
An equation of the line tangent to the graph of g at x = 3 is y = 5x + 4. Find g(3) and g′(3).
Verified step by step guidance1
Identify that the equation of the tangent line is given as y = 5x + 4, which is in the form y = mx + b, where m is the slope of the tangent line.
Recognize that the slope of the tangent line, m, is equal to the derivative of the function g at the point x = 3, so g'(3) = 5.
Understand that the point of tangency (x, y) on the graph of g is also a point on the tangent line. Since the tangent line is y = 5x + 4, substitute x = 3 into this equation to find the y-coordinate.
Calculate the y-coordinate by substituting x = 3 into the equation y = 5x + 4, which gives y = 5(3) + 4.
Conclude that the point (3, y) is on the graph of g, so g(3) is equal to the y-coordinate found in the previous step.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Line
A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which is equivalent to the derivative of the function.
Recommended video:
05:13
Slopes of Tangent Lines
Derivative
The derivative of a function at a specific point quantifies how the function's output changes as its input changes. It is denoted as g′(x) and can be interpreted as the slope of the tangent line to the graph of the function at that point. In this case, g′(3) is the slope of the tangent line at x = 3.
Recommended video:
05:44Derivatives
Function Value
The function value g(3) represents the output of the function g when the input is 3. It is the y-coordinate of the point on the graph of g where x equals 3. This value can be determined by substituting x = 3 into the function g, which is related to the tangent line equation provided.
Recommended video:
06:37Average Value of a Function
Related Practice
Textbook Question
The volume V of a sphere of radius r changes over time t.
c. At what rate is the radius changing if the volume increases at 10 in³ when the radius is 5 inches?
Textbook Question
Find the derivative the following ways:
Using the Product Rule or the Quotient Rule. Simplify your result.
f(x) = x(x-1)
Textbook Question
The volume V of a sphere of radius r changes over time t.
b. At what rate is the volume changing if the radius increases at 2 in/min when when the radius is 4 inches?
Textbook Question
Find the derivative the following ways:
Using the Product Rule or the Quotient Rule. Simplify your result.
g(t) = (t + 1)(t² - t + 1)
Textbook Question
If h(1) = 2 and h′(1) = 3, find an equation of the line tangent to the graph of h at x = 1.